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This Chalk Talk highlights applications of a discrete Gaussian in various areas, following a 2025 paper by Chinta, Jorgenson, Karlsson, and Smajlović. The topics include heat diffusion, Bessel functions, local limit laws, determinants of Laplacians, zeta functions, trigonometric sums, and volume formulas.

From de Moivre in 1773 onward, the Gaussian function has been central to several branches of mathematics and its applications. In a paper co-authored with Chinta, Jorgenson, and Smajlović (J. Phys. A, 2025), we argue that there exists a discrete analogue of the Gaussian that likewise has numerous applications. This Chalk Talk highlights several of these. It is worth noting that the Bessel function appearing in the discrete Gaussian is regarded here as a function of two variables, in contrast to the classical viewpoint.

We prove a discrete local limit theorem that may offer advantages for practitioners when approximating small sums of integer-valued random variables. In particular, it eliminates the need for continuity corrections, since it defines a probability distribution directly on the integers.

We obtained an asymptotic expansion for the determinants of lattice Laplacians in all dimensions. In the physics literature, only the leading growth rate had been determined in all dimensions, while the more complete asymptotics were previously known only in two dimensions.

In collaboration with Friedli and Müller, we showed that the discrete Gaussian naturally gives rise to zeta functions that approximate number-theoretic zeta and L-functions. In particular, our perspective explains why known special values, as well as certain trigonometric sums appearing in physics, turn out to be integral—or rational multiples of a power of π—by revealing their underlying combinatorial origin.

Together with Pallich, we observed that the centuries-old volume formulas for spheres can be reinterpreted in terms of discrete zeta values. Our expression is analogous to the volume formulas for arithmetic quotients discovered by Minkowski, Siegel, Weil, Langlands, and Harder.